Tensor networks-a new tool for old problems

A new renormalization group approach that maps lattice problems to tensor networks may hold the key to solving seemingly intractable models of strongly correlated systems in any dimension. A Physics Viewpoint on arXiv:0903.1069Comment: 5 pages, 2 figure

The landscape of the Hubbard model

I present a pedagogical survey of a variety of quantum phases of the Hubbard model. The honeycomb lattice model has a conformal field theory connecting the semi-metal to the insulator with Neel order. States with fractionalized excitations are linked to the deconfined phases of gauge theories. I also consider the confining phases of such gauge theories, and show how Berry phases of monopoles induce valence bond solid order. The triangular lattice model can display a metal-insulator transition from a Fermi liquid to a deconfined spin liquid, and I describe the theory of this transition. The bilayer triangular lattice is used to illustrate another compressible metallic phase, the `fractionalized Fermi liquid'. I make numerous connections of these phases and critical points to the AdS/CFT correspondence. In particular, I argue that two recent holographic constructions connect respectively to the Fermi liquid and fractionalized Fermi liquid phases.Comment: 56 pages, 16 figures; TASI and Chandrasekhar lectures; (v3) expanded discussion of phases with Fermi surfaces; (v5) added section on Mott transition on the triangular lattic

Quantum phase transitions of antiferromagnets and the cuprate superconductors

I begin with a proposed global phase diagram of the cuprate superconductors as a function of carrier concentration, magnetic field, and temperature, and highlight its connection to numerous recent experiments. The phase diagram is then used as a point of departure for a pedagogical review of various quantum phases and phase transitions of insulators, superconductors, and metals. The bond operator method is used to describe the transition of dimerized antiferromagnetic insulators between magnetically ordered states and spin-gap states. The Schwinger boson method is applied to frustrated square lattice antiferromagnets: phase diagrams containing collinear and spirally ordered magnetic states, Z_2 spin liquids, and valence bond solids are presented, and described by an effective gauge theory of spinons. Insights from these theories of insulators are then applied to a variety of symmetry breaking transitions in d-wave superconductors. The latter systems also contain fermionic quasiparticles with a massless Dirac spectrum, and their influence on the order parameter fluctuations and quantum criticality is carefully discussed. I conclude with an introduction to strong coupling problems associated with symmetry breaking transitions in two-dimensional metals, where the order parameter fluctuations couple to a gapless line of fermionic excitations along the Fermi surface.Comment: 49 pages, 19 figures; Lectures at the Les Houches School on "Modern theories of correlated electron systems", France, May 2009; and at the Mahabaleshwar Condensed Matter School, International Center for Theoretical Sciences, India, Dec 2009; (v2) expanded introductory discussion of cuprate phase diagram; (v2) corrected typo

Finite temperature dissipation and transport near quantum critical points

I review a variety of model systems and their quantum critical points, motivated by recent experimental and theoretical developments. These are used to present a general discussion of the non-zero temperature crossovers in the vicinity of a quantum critical point. Insights are drawn from the exact solutions of quantum critical transport obtained from the AdS/CFT correspondence. I conclude with a discussion of the role of quantum criticality in the phase diagram of the cuprate superconductorsComment: 27 pages, 10 figures, Contributed chapter to the book "Understanding Quantum Phase Transitions," edited by Lincoln D. Carr (Taylor & Francis, Boca Raton, 2010

Universal low temperature theory of charged black holes with AdS2_2 horizons

We consider the low temperature quantum theory of a charged black hole of zero temperature horizon radius $R_h$, in a spacetime which is asymptotically AdS$_{D}$ ($D > 3$) far from the horizon. At temperatures $T \ll 1/R_h$, the near-horizon geometry is AdS$_2$, and the black hole is described by a universal 0+1 dimensional effective quantum theory of time diffeomorphisms with a Schwarzian action, and a phase mode conjugate to the U(1) charge. We obtain this universal 0+1 dimensional effective theory starting from the full $D$-dimensional Einstein-Maxwell theory, while keeping quantitative track of the couplings. The couplings of the effective theory are found to be in agreement with those expected from the thermodynamics of the $D$-dimensional black hole.Comment: 24 pages, 2 figures; based on part of my lectures at the 36th Jerusalem Winter School at the Israel Institute for Advanced Studie

The quantum phases of matter

I present a selective survey of the phases of quantum matter with varieties of many-particle quantum entanglement. I classify the phases as gapped, conformal, or compressible quantum matter. Gapped quantum matter is illustrated by a simple discussion of the Z_2 spin liquid, and connections are made to topological field theories. I discuss how conformal matter is realized at quantum critical points of realistic lattice models, and make connections to a number of experimental systems. Recent progress in our understanding of compressible quantum phases which are not Fermi liquids is summarized. Finally, I discuss how the strongly-coupled phases of quantum matter may be described by gauge-gravity duality. The structure of the large N limit of SU(N) gauge theory, coupled to adjoint fermion matter at non-zero density, suggests aspects of gravitational duals of compressible quantum matter.Comment: 35 pages, 21 figures; Rapporteur presentation at the 25th Solvay Conference on Physics, "The Theory of the Quantum World", Brussels, Oct 2011; (v2+v3+v4) expanded holographic discussion and referencin

Quantum Phases of the Shraiman-Siggia Model

We examine phases of the Shraiman-Siggia model of lightly-doped, square lattice quantum antiferromagnets in a self-consistent, two-loop, interacting magnon analysis. We find magnetically-ordered and quantum-disordered phases both with and without incommensurate spin correlations. The quantum disordered phases have a pseudo-gap in the spin excitation spectrum. The quantum transition between the magnetically ordered and commensurate quantum-disordered phases is argued to have the dynamic critical exponent $z=1$ and the same leading critical behavior as the disordering transition in the pure $O(3)$ sigma model. The relationship to experiments on the doped cuprates is discussed.Comment: 16 pages, REVTEX 3.0, 6 uuencoded Postscript figures appended, YCTP-xxx